A Trajectory Minimizing the Exposure of a Moving Object

A corridor $Y$ for the motion of an object is given in the space $X=\mathbb{R}^N$ ($N=2,3$). A finite number of emitters $s_i$ with fixed convex radiation cones $K(s_i)$ are located outside the corridor. The intensity of radiation $F(y)$, $y>0$, satisfies the condition $F(y)\ge \lambda F (\lambda y)$ for $y>0$ and $\lambda >1$. It is required to find a trajectory minimizing the value
$$ J(\cal T)=\sum_{i}\int\limits_{0}^1 F\big(\|s_i-t(\tau)\|\big)\,d\tau $$
in the class of uniform motion trajectories $\cal T=\big\{ t(\tau)\colon 0\le \tau\le 1,\ t(0)=t_*,\ t(1)=t^*\big\}\subset Y$, $t_*,t^*\in \partial Y$, $t_*\ne t^*$. We propose methods for the approximate construction of optimal trajectories in the case where the multiplicity of covering the corridor $Y$ with the cones $K(s_i)$ is at most 2.